Example 3.4 We take the graph \(G\)  the circulant graph on \(\mathbb {Z}_{18}\)  with generating set \(S=\lbrace 1,2,4,6,7,8,10,11,12,14,16,17\rbrace \) . Here, \(S_1\)  is \(\lbrace 1,2,4,6,12,14,16,17\rbrace \) . We first give a 9-total coloring of the subgraph induced by \(S_1\)  . To give a total 9-coloring, we use the total color matrix of the canonical coloring of a complete graph of order 9. This gives us the starting colors for the subdiagonals starting at \(1,2,4,6\)  to be \(6,2,3,4\) . correspondingly the subdiagonals starting at \(12,14,16,17\)  to be \(7,8,9,5\)  .The part of graph induced by \(\lbrace 7,8,12,13\rbrace \) , which is also a circulant graph, can be given a 4 coloring ( from Corollary 2.3.1 of [1]}). This gives us a 13 total coloring of the graph which is also equitable. Now, we take the rainbow Hamiltonian cycle induced by the set \(\lbrace 1,19\rbrace \)  and give it a 2- coloring using two extra colors (replacing the original coloring), which is possible as the cycle induced is Hamiltonian (and hence even). This gives us a neighborhood sum distinguishing total coloring using 15 colors as the colors replaced in each row and/or column happens to be having a distinct sum owing to the rainbow nature of the prior coloring. The total color matrix of the subgraph corresponding to \(S_1\)  is given in Table 5.
